Steady film flow over a substrate with rectangular trenches forming air inclusions

Film flow along an inclined, solid substrate featuring periodic rectangular trenches may either completely wet the trench floor (Wenzel state) or get pinned on the entrance and exit corners of the trench (Cassie state) or assume other configurations in between these two extremes. Such intermediate configurations are examined in the present study. They are bounded by a second gas-liquid interface inside the trench, which adheres to its walls forming two three-phase contact lines, and encloses a different amount of air under different physical conditions. The Galerkin finite-element method is used to solve the Navier-Stokes equations in a physical domain, which is adaptively remeshed. Multiple steady solutions, connected by turning points and transcritical bifurcations as well as isolated solution branches, are revealed by pseudo-arc-length continuation. Two possible configurations of a single air inclusion inside the trench are examined: the inclusion either surrounds the upstream convex corner or is attached to the upstream trench wall. The penetration of the liquid inside the trench is enhanced primarily by increasing either the wettability of the substrate or capillary over viscous forces or by decreasing the flow rate. Flow hysteresis may occur when the liquid wetting of the upstream wall decreases abruptly, leading to drastically different flow patterns for the same parameter values. The interplay of inertia, viscous, gravity, and capillary forces along with substrate wettability determines the volume of the air encapsulated in the trench and the extent of deformation of the outer free surface.

Figure 1

Sketch of the film flow over a rectangle trench with air inclusion at the upstream corner (left) or at the upstream wall (right). The geometric parameters, the unit vectors on each surface and the contact points and angles are indicated. When the indicated lengths are dimensional, the film thickness in the entrance is $H$; when they are dimensionless, it is unity.

Figure 2

Part of the mesh around the trench in the physical domain before (left) and after the remeshing (right). Here the two contact points are on the upstream side wall and the bottom of the trench.

Figure 3

Characteristic measures of film deformation.

Figure 4

Comparison of transient VOF simulations in OpenFoam by Lampropoulos et al. (a) with our predictions (b) for $\mathrm{Re}=5.5\times {10}^{-4},\mathrm{Ka}=0.139,\theta ={30}^{\circ },w=14$, and $d=12.2$.

Figure 5

Dependence of the pressure inside the air inclusion on the volume of air inside the air inclusion for different values of the Reynolds number, $\mathrm{Ka}=2,\theta ={60}^{\circ },d=5$, and $w=5$.

Figure 6

Close-up views around the trench and just before and after it showing the streamlines and the pressure field, respectively, for two values of Reynolds number: (a), (b) $\mathrm{Re}=1$, (c), (d) $\mathrm{Re}=3$ at parameter set $\mathrm{Ka}=2,\theta ={60}^{\circ },w=5$, and $d=5$, with $l=30$.

Figure 7

Map of the steady-state solutions in terms of the wetting of contact line A $\left({\mathrm{CL}}_A\right)$, contact line B $\left({\mathrm{CL}}_B\right)$, or the deformation amplitude $\left(A_d\right)$ for $\mathrm{Ka}=2,\theta ={60}^{\circ },d=5$, and $w=5$. Here and in similar figures that follow the symbols in the solution family correspond to one of the flow patterns around the three figures.

Figure 8

Enlarged capillary-gravity region for $\mathrm{Ka}=2,\theta ={60}^{\circ },d=5$, and $w=5$ along with contours of the pressure field for $\mathrm{Re}=0.18$ and $\mathrm{Re}=0.7$.

Figure 9

Enlarged transition zone (lower branch) for $\mathrm{Ka}=2,\theta ={60}^{\circ },d=5$, and $w=5$ along with contours of the stream function for $\mathrm{Re}=2$ and $\mathrm{Re}=3.8$.

Figure 10

Enlarged inertia-gravity zone (upper branch) for $\mathrm{Ka}=2,\theta ={60}^{\circ },d=5$, and $w=5$ along with contours of the pressure field for $\mathrm{Re}=7$ and $\mathrm{Re}=9.8$.

Figure 11

Solution families for the wetting by contact line A $\left({\mathrm{CL}}_A\right)$ or the deformation amplitude $\left(A_d\right)$ for $\mathrm{Ka}=2,d=5$, and $w=5$ under different contact angles, along with certain film configurations.

Figure 12

Solution families for the wetting by contact line A $\left({\mathrm{CL}}_A\right)$ for $\theta ={60}^{\circ },d=5$, and $w=5$ for different liquids characterized by their $\mathrm{Ka}$.

Figure 13

Solution families for the wetting by contact line A $\left({\mathrm{CL}}_A\right)$ or amplitude of deformation $\left(A_d\right)$ for $\mathrm{Ka}=6,\theta ={60}^{\circ },d=5$, and $w=5$ along with film shapes and contours of the stream function.

Figure 14

Solution families in terms of ${\mathrm{CL}}_A$ as a function of (a) ${\mathrm{Bo}}_m$ and (b) We, for $\theta ={60}^{\circ },d=5,w=5$ for different liquids. The lines correspond to the same $\mathrm{Ka}$ as in Fig. 12.

Figure 15

Solution families in terms of ${\mathrm{CL}}_A$ or $A_d$, for $\mathrm{Ka}=2,d=5$, and $w=5$ under different inclination angles along with film configurations.

Figure 16

Solution families in terms of wetting by contact line A $\left({\mathrm{CL}}_A\right)$ or amplitude of deformation $\left(A_d\right)$ for $\mathrm{Ka}=2,\theta ={60}^{\circ }$, and $w=5$ and different trench depths.

Figure 17

Solution families in terms of the wetting by contact line A $\left({\mathrm{CL}}_A\right)$ or amplitude of deformation $\left(A_d\right)$ for $\mathrm{Ka}=2,\theta ={60}^{\circ }$, and $d=5$ and different trench widths.

Figure 18

Solution family in terms of ${\mathrm{CL}}_A$ or $A_d$, for $\mathrm{Ka}=2,d=5,w=5$, and $\theta ={60}^{\circ }$ along with film shapes and contours of the streamlines.